Target Zones and Exchange Rate Dynamics

The paper addresses how an exchange rate should behave inside a target zone – a band within which governments commit to keep the exchange rate, intervening only at the edges – a question the author says “little research has been done on” despite the expected shift of the world monetary system “from one of freely floating rates to one in which governments commit themselves to keep exchange rates within broad target zones.” A naive view might hold that the exchange rate behaves as if freely floating until it reaches the edge of the band, then acts like a fixed rate. The paper argues this “cannot be right” given forward-looking speculators: the expectation that the rate will be stabilized if it moves too far should affect its behavior well before the band is actually reached or defended (Introduction). The model presented is explicitly a cleaner, more tractable restatement of the model in Krugman’s own earlier unpublished paper (Krugman 1987), using a continuous-time formulation of the fundamentals process and a different approach to determining where the exchange rate reaches the edge of the band.

The exchange rate is modeled as determined by two factors: a “fundamentals” variable that evolves exogenously over time, and the exchange rate’s own expected rate of change. The paper is deliberately agnostic about what the fundamentals represent – they “might be monetary variables, or… a variety of other factors” – and assumes only that the (log) fundamental level x, which would be the exchange rate if no appreciation or depreciation were expected, follows a driftless diffusion process, dx = σdz (The Model, Eq. 1). The non-intervention (“shadow”) exchange rate s is related to fundamentals and to expected changes in the rate by s = x + E[ds/dt] (Eq. 2); the actual exchange rate is assumed capped at the band’s upper limit and floored at its lower limit, with the authorities able to ensure the rate never exceeds these limits however that intervention is actually implemented, and units are chosen so the band is symmetric around zero.

Inside the band, where the actual and shadow exchange rates coincide, the author solves for an equilibrium function s = g(x) satisfying the model’s differential equation and shows that the general solution has the form s = x + A(e^(rx) - e^(-rx)), where r is determined by the volatility of fundamentals and the sensitivity of the exchange rate to its expected rate of change, and where symmetry of the band around zero pins down the functional form. The paper demonstrates that in a target zone, the constant A must be negative, which makes g(x) an S-shaped curve: flatter than the 45-degree line, lying below the 45-degree line in the upper part of the band and above it in the lower part. The geometric intuition is Jensen’s-inequality-based: because the expected change in fundamentals is zero, the expected change in the exchange rate (a function of fundamentals) is positive wherever the fundamentals-exchange-rate relationship is convex and negative wherever it is concave; in the lower part of the band this “drags” the rate above the 45-degree line, and in the upper part it drags the rate below the fundamental level (Exchange Rate Behavior Inside the Band).

The paper shows, by ruling out alternative geometries, that the equilibrium function g(x) must be precisely tangent to the top of the band (and, by symmetry, to the bottom) – not merely crossing it at an angle. Considering the possible ways g(x) could intersect the band’s upper limit – a convex g(x) crossing it, a concave g(x) crossing it, or a concave g(x) just tangent to it – the author argues that in the two crossing cases the actual exchange rate path would be “dragged back inside the band” before reaching the point at which intervention is assumed to become necessary, contradicting that assumption; “only a tangency equilibrium is possible.” This tangency result is then used to derive a simple closed-form measure of the target zone’s stabilizing effect.

Exploiting the fact that g(x) reaches its maximum exactly at the upper edge of the band (where its slope equals one), the paper derives x̄ − s̄ = 1/r, a simple closed-form expression for how much further the fundamentals can rise, relative to a freely floating rate, before the exchange rate would reach the top of the band. This quantity – the gap between the fundamental level at which intervention becomes necessary and the exchange rate observed at the edge of the band – directly measures the extent to which the target zone’s mere existence widens the range of fundamentals compatible with a given exchange rate, even when the authorities are not currently intervening. The paper states that this stabilizing effect “depends on two parameters: [the sensitivity of the exchange rate to the expected rate of depreciation], and… the volatility of the fundamentals” (The Stabilizing Effect of the Target Zone).

Because the market recognizes that the authorities will defend the band if necessary in the future, the exchange rate near the top of the band has “more room to go down than up” (and conversely near the bottom), and this asymmetric realization itself creates regressive expectations that stabilize the rate before any actual intervention occurs. This is the paper’s central qualitative claim and the reason the S-curve lies everywhere flatter than the 45-degree line that a freely floating exchange rate would trace against the same fundamentals: fluctuations in fundamentals are “less than fully reflected in the exchange rate” under a target zone (The Stabilizing Effect of the Target Zone).

The paper extends the analysis to the case where the market believes the authorities will defend the zone only with probability φ (and with probability 1−φ no response will occur when the rate reaches the edge), so that reaching the edge becomes “a testing of the authorities’ resolve”: if they defend, the rate moves onto the full-credibility schedule g(x); if not, the system reverts to free floating at the 45-degree line. The no-arbitrage requirement that the expected jump at the testing point equal zero (since an expected jump would otherwise imply an infinite expected rate of appreciation) pins down a modified function, denoted g(x,φ), which intersects the edge of the band at a fundamentals level below the full-credibility intersection point, making it steeper – and therefore less stabilizing – than g(x). As credibility φ goes to one, g(x,φ) converges to the full-credibility function g(x); as φ goes to zero, it converges to the 45-degree free-floating line (Dynamics with Imperfect Credibility).

The paper is explicit that it omits two of the usual sources of interesting exchange-rate dynamics: because fundamentals follow a pure random walk, there is “no room for responses to anticipated future shocks,” and the model “lacks any ‘intrinsic’ dynamics arising from slowly adjusting variables such as the distribution of world wealth or a sticky domestic price level.” The payoff to excluding these features, the author states, is that it isolates the impact of the target zone itself on exchange-rate dynamics, allowing the paper to “derive some surprisingly clear and powerful conclusions about this impact” (The Model). The paper’s own conclusion section acknowledges that making the model “more realistic and interesting” – via a more complex fundamentals process and intrinsic dynamics such as sluggish price adjustment – is “the obvious next step” for this research.

The paper’s stated main result is that “the expectation that the authorities will defend a target zone will exert a stabilizing influence on exchange rate behavior inside the zone,” with the extent of that stabilization depending on the sensitivity of the exchange rate to its own expectations, the volatility of the underlying determinants of the exchange rate, and the credibility of the authorities’ commitment (Conclusions). The author also notes that the method of solution – in which the convexity of the relationship between fundamentals and the exchange rate plays the central role – was, at the time of writing, “still somewhat unfamiliar in international economics.”